Let f:R--R be a bounded function. Prove that f is continuous if and only if the graph of f is a closed subset of R^2. What if f is unbounded?
Please, help

Question

Let f:R-->R be a bounded function. Prove that f is continuous if and only if the graph of f is a closed subset of R^2. What if f is unbounded? 
Please, help

loser66 · Answer

@Miracrown

loser66 · Answer

It's advance calculus part compact sets

anonymous · Answer

Why don't we start with the first direction.

anonymous · Answer

We will assume that $f$ is continuous and we want to show that the graph of $f$ is closed in $\mathbb{R}^2$.

anonymous · Answer

To show that it is closed we will prove that the complement of the graph is open.

anonymous · Answer

We will consider an arbitrary point in the complement and show that we can find a ball around the point contained in the complement.

loser66 · Answer

Show me more, please. I have no clue. :)

anonymous · Answer

Do you understand where I'm going with this so far?

anonymous · Answer

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